What the heck is Mana?

Mana is the mystical energy resource a spell caster draws upon when using spells. It is similar in some ways to the rage a warrior generates before combat, the energy generated by a rogue before his kidney shot or the focus a hunter's pet gains as it charges. The amount of mana available to the caster is shown on the player GUI with a blue bar directly below their health bar. On character creation, all casters start with a full mana bar. This bar will drop as spells are used.

Mana is determined by your class level and your Intellect. Base Mana is a term used to refer how much mana a character has before it is modified by Intellect and other Item Effects. Many spells require a percentage of your base mana to be cast.

How do I get mana back?

Characters regenerate both health and mana based on the total amount of Spirit on their character, as well as Talents that can aid in the raising of regeneration rate independently. Mana regen while casting is greatly reduced as opposed to resting (eating, drinking in a non combat state), but can also be regained instantly in and out of combat by using potions from alchemists or other magical means.
As a Mage, you can conjure water that will restore mana while sitting and drinking, and you can share with friends. Mages can also conjure stones that restore mana and are used much like a potion, but cannot be shared with others.